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Problem 4

Show that the low-temperature specific heat of a free electron gas
with exchange goes as T/ln(T).

(Note: the expression given in class for the Hartree-Fock energy
inadvertently left out the absolute value symbol enclosing what's inside 
the log; see e.g. Ashcroft & Mermin, Eqs. (17.19),(17.20)).

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Problem 5:  

Repeat the calculation of Wigner and Seitz (Phys.Rev. 43, 804 (1933)) for
metallic Sodium. In particular, make a plot of the wave functions and find the 
radii for energies E=0.5, 0.55 and 0.6 Rydbergs. Verify the WS results for
lattice constant, binding energy and compressibility of Na.

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Problem 6: 
(reference: Koster and Slater, Phys.Rev. 94, 1498 (1954))

Consider a two-dimensional square lattice with atomic px, py orbitals
at every lattice site. Assume orbitals at different sites are orthogonal.

(a) List the non-vanishing independent Hamiltonian matrix elements Enn'(R),
up to next-nearest-neighbor sites, and discuss their sign and relative magnitude
qualitatively.

(b) Construct Enn'(k)

(c) Construct the band structure in the gamma-X, X-L and gamma-L directions
and draw E versus k qualitatively. You should pay attention to degeneracies, 
ordering of energies at symmetry points, sign of the slope of En(k) and relative
bandwidths (qualitatively) using the results of (a).

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